curl-Yamabe问题
The curl-Yamabe problem
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中文总结 AI 辅助
本文引入curl-Yamabe问题，定义共形不变常数，证明其上界及严格不等式条件，并得到极小元存在性。
中文摘要 AI 辅助
我们引入并研究了一个与curl算子相关的Yamabe型变分问题，该问题定义在闭定向黎曼流形$(M^n,g)$上的中阶微分形式上，其中$n\equiv3\pmod4$。相应的curl--Yamabe常数通过最小化一个由$\\|{\rm curl}\\,\alpha\\|_{\frac{2n}{n+1}}$和$\int_M\langle{\rm curl}\\,\alpha,\alpha\rangle$构成的共形不变、规范不变的商来定义，其Euler--Lagrange方程是临界curl--Yamabe方程${\rm curl}\\,\alpha=\frac{n+1}{2}|\alpha|^{\frac{2}{n-1}}\alpha$。利用球面上的尖锐curl--Sobolev不等式和Aubin型截断构造，我们证明了对于每个这样的流形，$0<Y_{\rm curl}(M,[g])\le Y_{\rm curl}(\mathbb{S}^n)$，并且当$Y_{\rm curl}(M,[g])<Y_{\rm curl}(\mathbb{S}^n)$时，下确界可以达到。此外，对于$n>3$，我们通过检测Weyl张量的测试形式展开，建立了非局部共形平坦流形的严格不等式，并推导出求解curl--Yamabe方程的极小元的存在性。
英文摘要
We introduce and study a Yamabe-type variational problem associated with the curl operator on middle-degree forms on a closed oriented Riemannian manifold $(M^n,g)$ with $n\equiv3\pmod4$. The corresponding curl--Yamabe constant is defined by minimizing a conformally invariant, gauge-invariant quotient built from $\|{\rm curl}\,α\|_{\frac{2n}{n+1}}$ and $\int_M\langle{\rm curl}\,α,α\rangle$, and its Euler--Lagrange equation is the critical curl--Yamabe equation ${\rm curl}\,α=\frac{n+1}{2}|α|^{\frac{2}{n-1}}α$. Using sharp curl--Sobolev inequalities on the sphere and an Aubin-type cut-off construction, we prove that $0<Y_{\rm curl}(M,[g])\le Y_{\rm curl}(\mathbb{S}^n)$ for every such manifold, and that the infimum is attained whenever $Y_{\rm curl}(M,[g])<Y_{\rm curl}(\mathbb{S}^n)$. Moreover, for $n>3$ we establish the strict inequality for manifolds that are not locally conformally flat, via a test-form expansion detecting the Weyl tensor, and deduce existence of a minimizer solving the curl--Yamabe equation.
发表机构
- Albert-Ludwigs-Universität Freiburg（弗莱堡大学）
- Wuhan University, School of Mathematics and Statistics（武汉大学数学与统计学院）
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